% Copyright (C) 2001 Michel Juillard%% computes second order partial derivatives% uses Abramowitz and Stegun (1965) formulas 25.3.24 and 25.3.27 p. 884function hessian_mat = hessian(func,x,gstep_,varargin)if ischar(func)==1 func = str2func(func);endn=size(x,1);%h1=max(abs(x),gstep_*ones(n,1))*eps^(1/3);h1=max(abs(x),sqrt(gstep_)*ones(n,1))*eps^(1/6);h_1=h1;xh1=x+h1;h1=xh1-x;xh1=x-h_1;h_1=x-xh1;xh1=x;f0=feval(func,x,varargin{:});disp(sprintf('Starting Log Posterior for Hessian=%10.5f',f0)); f1=zeros(size(f0,1),n);f_1=f1;for i=1:n    xh1(i)=x(i)+h1(i);    f1(:,i)=feval(func,xh1,varargin{:});    xh1(i)=x(i)-h_1(i);    f_1(:,i)=feval(func,xh1,varargin{:});    xh1(i)=x(i);    i=i+1;endxh_1=xh1;hessian_mat = zeros(size(f0,1),n*n);for i=1:n    if i > 1        k=[i:n:n*(i-1)];        hessian_mat(:,(i-1)*n+1:(i-1)*n+i-1)=hessian_mat(:,k);    end     hessian_mat(:,(i-1)*n+i)=(f1(:,i)+f_1(:,i)-2*f0)./(h1(i)*h_1(i));    temp=f1+f_1-f0*ones(1,n);        for j=i+1:n        xh1(i)=x(i)+h1(i);        xh1(j)=x(j)+h_1(j);        xh_1(i)=x(i)-h1(i);        xh_1(j)=x(j)-h_1(j);        hessian_mat(:,(i-1)*n+j)=-(-feval(func,xh1,varargin{:})-feval(func,xh_1,varargin{:})+temp(:,i)+temp(:,j))./(2*h1(i)*h_1(j));        xh1(i)=x(i);        xh1(j)=x(j);        xh_1(i)=x(i);        xh_1(j)=x(j);        j=j+1;    end    i=i+1;end% 11/25/03 SA Created from Hessian_sparse (removed sparse)